This is a really interesting question!
So what would it feel like to squeeze this orange-sized ball?
So, assuming that the earth keeps most of it's material properties after being shrunk, and you hold it the moment it was transformed then it won't be comfortable experience.
The Weight
For one, as you pointed out the average density of the earth is about $4g/cm^3$ (Wolfram gives $5.515 g/cm^3$). Going by this table the most common metal that is close to that number is cast iron. So it would feel a little lighter than an iron ball.
The Texture
According to this "The major, intermediate and minor diameters of [a] grade 2 orange [is] 84.1, 77.4 and 75.5 mm, respectively". This means that the diameters of an orange deviate a lot between axes. That is not the case with the earth. According to Wikipedia the earth's radius ranges from 6,378 kilometers at the equator and 6,357, so the earth is a lot closer to a sphere than our orange. Furthermore, as I am sure you've heard, the earth's crust is very smooth. A bad astronomy post breaks down this fact by proving that the earth is in fact smoother than a standardized billiard ball.
The Burn
Of course none of this compares with the fact that as soon as you pick up the orange sized planet you will feel just how hot the earth is. According to Wikipedia the temperature of the mantle ranges from 200 centigrade to 4000 centigrade. For the most part we are shielded from those temperatures by the 30 km crust right above it. But in our orange sized earth that crust has now been shrunk to about 0.3-0.5 mm thick.
Using the formula for heat conduction through spherical shells, and assuming the inner radius is 80mm, the outer radius is 80.5mm, a $k$ value of 1.3 W/mK and you are at r.t.p we get:
$$\frac{\delta Q}{\delta t} = 4k\pi\frac{(T_1 - T_2)r_1r_2}{r_2 - r_1} \approx 38kW$$
so even though at first the "crust" would be relatively cool, it would almost instantly heat up to the point where it would be unbearable. Also, this was done with the 200 centigrade figure but in reality it would feel much much hotter.
The Squeeze
But what about the real reason we came here for? If you were wearing industrial heat resistant gloves that also felt like they weren't there at all, how would the orange sized earth "give"? Well for one as you pointed out the reason the mantle close to our crust is so hot is partly due to the fact that it convects. Its almost fluid behaviour means that temperature can circulate.
But a lot of that is due to the behaviour of rock and other materials as they are heated and compressed with insane pressure. In actuality, the mantle is mostly composed of solid rock that is under so much pressure and heat that it behaves somewhat like a fluid.
However, that is no longer the case once you have shrunk the Earth to the size of an orange. The core might be primarily liquid iron and nickel but you wont be able to really squeeze it like a water balloon.
That being said, the temperature of the inner parts of the mantle as well as the core are way above the melting point for the materials found there so the inner portions would quickly melt.
If anything I think it would feel more like a really hot and tough bonbon or jawbreaker with a liquid core. I also speculate that it could expand very rapidly due to the large change in pressure from the sudden shrinking.
What about making ourselves big?
Would the physics feel significantly different?
YES. For one, there exist the problem of making objects big when they are only feasible at smaller scales. The only time when an average person is concerned with gravity is when they have to deal with Earth's influence. However, once you are at the planetary scale you find that you will inevitably have to deal with your own gravity.
If you look at photos of "small" astronomical objects like asteroids and comets then you will see that their shapes vary a lot from case to case. However, the bigger you get, the more these objects seem to approach a sphere-like structure. This can be (very crudely and simplistically) explained as tendency to for these systems to minimize they're potential energy. For a system that is experiencing its own gravity, this is a sphere, since it is the shape where the most mass can be closest to the center.
As an aside, Randall Munroe, the author of XKCD comics, has another series in which he explores the types of questions and in several he has to deal with scaling up objects and their effects. I really recommend this series as it is full of interesting questions like yours. In particular I would look at the earth sized bowling ball.
So what? You are a ball now, what about the squeeze?
Well it just so happens that you are not the only entity that has to deal with your gravity. Earth now has to come to terms that you are now much larger and massive than it. If we scale ourselves up so that the Earth (about $1\times10^{21} \ m^3$) is the size of an orange relative to us then going by the dimensions of that grade 2 orange:
$$V_{\text{orange}} = \frac{4}{3}\pi abc \approx 2.06 \times 10^{-3} m^3$$
Using the fact that mass generally scales up with volume we can use proportions. Assuming you are around 70kg:
$$M_{\text{big-you}} = \frac{V_{\text{earth}}}{V_{\text{orange}}}M_{\text{small-you}} \approx 3.4 \times 10^{25} kg$$
which, according to wolfram, happens to be almost 6 times the mass of Earth. So the earth would now orbit you. As fun as that is I would not recommend to try to squeeze since tidal forces are now certain to have an effect between you two.
For one, assuming you are a sphere of average human density (about the density of water) your Roche limit is now:
$$d = R_{\text{earth}}\left(2\frac{M_{\text{big-you}}}{M_{\text{earth}}}\right)^{\frac{1}{3}} \approx 1.4 \times 10^7 m$$
which according to wolfram is about the diameter of the Earth. So you must keep a distance of one Earth, from the Earth. If you don't you will both start breaking up into dust in spectacular fashion.
Finally, if you somehow manage to ignore all that and become Galactus then you could easily squeeze through the hard (now relatively cool) crust to find that the mantle is once again your limiter. It is essentially solid but very hot rock that "flows" but not at the rate to make it seem like a balloon or feel like anything you've touched before. What would be fun is that now that you have squeezed the crust some of that is now orbiting you! Like the Galactus equivalent of crumbs.
