I was able to find a copy of the book you read. It addresses length contraction in Chapter 9 (more precisely, Section 9-1) and time dilation in Chapter 10. I confirm that the strange rates of the "non-Einsteinian clocks" of Chapter 10 are calculated by Jefimenko neglecting length contraction, as reported in Laff70's answer. Therefore, I'll address only length contraction.
Here's a short excerpt from the introduction of Ch. 9:
There is a widespread belief that according to relativity theory the length of a body becomes shorter when the body moves. This is incorrect. The length of a body is defined as the length measured when the body is at rest relative to the observer and is an invariant quantity.
This claim is not in contradiction with relativistic length contraction, because the word length, as Jefimenko uses it, is what we commonly call proper length, which doesn't change as the body is put in motion. As the body moves with constant velocity, an observer in the object's rest frame would measure the proper length, while an observer moving with respect to it would measure a contracted length. Length contraction is a frame-dependent thing.
Another quote from Chapter 9, Section 9-1:
He [Einstein] also suggested the following method for measuring the length of a moving object (rod): observers in the stationary system ascertain at what points in the stationary system the two ends of the moving rod are located at the same time $t$; the distance between these two points is the "length of a moving object".
Yes, this is the procedure which people should have in mind when they talk about length contraction. Jefimenko then goes on to say:
Einstein's measuring procedure actually constituted the definition of a new quantity, which he called "length of a moving body," different from "length" in the conventional sense.
Only if you believe that "length in the conventional sense" is what we call "proper length". I think that a point could be made that also Einstein's definition could be called "length" in a sense very near to the common sense of observers in the stationary frame. Anyway, up to now, Jefimenko's departure from mainstream views on length contraction is only a matter of definitions, not substantial.
After that Jefimenko notices that it is not clear how the visual appearance of a moving body can be associated with Einstein's measuring procedure, because said appearance is an optical effect. This is true, but the particulars of this optical effect do have been worked out, see the Wikipedia page on Terrell rotation or this nice article by V. F. Weisskopf. If I understand it correctly, the resulting "distorted" shapes are precisely what Jefimenko later calls "retarded" shapes. Therefore, if the observers take into account the speed of light, they can compute the positions in which different points of the object are found at the same time $t$, and thus perform Einstein's procedure. They simply know the mechanism by which the object's visual appearance is optically "distorted" and can, in principle, undo this effect in their calculations.
After that, Jefimenko remarks that in Chapters 6 and 7 he used the retarded length and volume of moving charge distributions to derive the correct relativistic transformation equations, not the contracted length and volume. He accomplishes this by considering the fields produced by a moving distribution in a point. I'm not sure of what does he want to prove with this remark: it looks obvious to me that, in order to compute the electromagnetic fields in a point $\vec{x}$, you have to use retarded expressions of the charge and current distributions. This is because there is not instantaneous action at a distance, so you have to consider charges and currents as they were in the past, more so as more far these charges and currents are from $\vec{x}$. On the other hand, length contraction arises from considering the instantaneous positions of the charges at a given time, so it is clear that Lorentz contraction in itself shouldn't be relevant to this kind of calculations.
Then he claims that:
... the retarded field theory, without using Lorentz contraction for determining the effective shape of the moving charge, yields relativistically correct fields of the charge.
Indeed, he never explicitly used Lorentz contraction in his (correct) calculations. Nevertheless he used Lorentz transformations, of which Lorentz contraction is a consequence; therefore, he never had to take Lorentz contraction into account because he was using more fundamental facts. It follows that Lorentz contraction is implicit in a lot of passages he made. For example, he found that the moving charge distribution $\rho_m$ is connected to the static one $\rho_s$ by the relation $\rho_m = \gamma \rho_s$ (it is equation 6-3.4; I'm considering everything still in the rest frame of the charge distributions, this is why I took the current term to be zero). I'm saying that this implicitly involves Lorentz contraction because if the volume is contracted in a direction by a factor $\gamma$ (i.e., in that direction $L \rightarrow \frac{L}{\gamma}$), in order for the total charge to stay the same, its density has to be multiplied by the same factor $\gamma$, just like he found.
In conclusion, I agree with all the premises that Jefimenko makes, but I simply cannot see how they could imply that Lorentz contraction is a fiction. Nevertheless, near the end of Section 9-1 he claims that this discussion brings us to the conclusion that...
... Lorentz contraction does not exist.
I simply can't understand this conclusion.
Finally, let me briefly address Example 9-1.1, which should prove that (quoting) "Lorentz contraction is not a true physical effect".
He shows that integrating Heaviside's formula for the electric field of a moving point charge along a line gives the electric field produced by a moving line charge, and this result is the same as that obtained on the basis of electromagnetic retardation (his Section 4-3). Then he observes that the same result can be obtained by relativistic considerations (his Example 7-6.3), transforming the field from the rest frame of the line charge to the laboratory frame.
Finally, he observes that, on the other hand, integrating Heaviside's formula over a contracted line charge (with a charge density accordingly multiplied by $\gamma$) gives the wrong result. This should prove that Lorentz contraction is not real.
My understanding of this fact is that it simply doesn't make sense to consider a contracted line in this last integral:
The "retarded" calculations of Section 4-3 were carried out considering a line charge having length $L$ in the lab frame, not in its own rest frame. We would need to Lorentz-contract the line only when transforming its length from its rest frame to the lab frame. But $L$ is already its length in the lab frame, so contracting it would be plainly wrong. Rather, if we wanted to consider it in its rest frame, we would need to Lorentz-dilate it from $L$ to $\gamma L$. But since we are making all our calculations in the lab frame, Lorentz contraction is not called into question.
The relativistic considerations of Example 7-6.3, on the other hand, do involve transformations between the line's rest frame and lab frame. Then let's see how Jefimenko handles the transformation of the lengths and... surprise! Just after Eq. 7-6.24 he says: "we transform $E_x'$, $R'$, $\lambda '$ and $L'$". A little later, he says that "$L'$ transforms like $x'$". This transformation, even if Jefimenko doesn't explicitly remark it, results precisely in a Lorentz contraction!
Finally, the integral of Heaviside's formula along the contracted line gives a wrong result because $L$ and $\lambda$ are already the length and density of the line in the lab frame, not those in its rest frame.
Contraction only arises when switching from the rest frame to the lab frame, this is why it must not be used in this Example. Not because it's a fiction.
