Why is ${\partial^i}{\partial_i\phi}$ = ${\partial^i {\phi}}{\partial_i{\phi}}$? This notation can be found on page 254 of Victor Stenger's Comprehensible Cosmos and in David Tong's Lectures on QFT (Equation 2.4 http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf), and in 
EDIT: on page 254 of Stenger's Comprehensible Cosmos, the Lagrangian is written with a  ${\partial^i}{\partial_i\phi}$ instead of the usual ${\partial^i {\phi}}{\partial_i{\phi}}$ (that David Tong uses).
Why is ${\partial^i}{\partial_i\phi}$ = ${\partial^i {\phi}}{\partial_i{\phi}}$ in QFT ? This fact is used to calculate the Lagrange Equations of Motion (The Klein Gordon Equation) from the Lagrange Density for a Scalar Field. 
This clearly isn't true for elementary functions like $y^2$ because ${\partial_y}{\partial_y\ ({y^2})}$ =/= $ {\partial_y {y^2}}{\partial_y{y^2}}$
 A: You have overlooked a letter. The kinetic term for the Klein-Gordon field is usually written as
$$ {\mathcal L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi. $$
However, the equations of motion (and the action, assuming vanishing of the fields at infinity) don't change if we add a total derivative (or divergence) to the Lagrangian. So we may subtract
$${\mathcal L}\to {\mathcal L}' = {\mathcal L} - \partial_\mu \left(\frac 12\phi \partial^\mu \phi\right). $$
Using the Leibniz rule (i.e. $(uv)'=u'v+uv'$), this modified Lagrangian is easily seen to be
$$ {\mathcal L}' = -\frac{1}{2} \phi \cdot \partial_\mu \partial^\mu \phi $$
which is essentially what you wrote except that you omitted the factor of $-\phi$.
A: OK just to be clear the lagrangian for a scalar field theory for scalar field is written as 
$ \mathcal{L} = \frac{1}{2} \partial^\mu \phi \partial_\mu \phi -\frac{1}{2} m^2 \phi^2  $
sometimes this is written as 
$ \mathcal{L} = \frac{1}{2} ( \partial^\mu \phi)^2 -\frac{1}{2} m^2 \phi^2  $
where 
$( \partial^\mu \phi)^2 = \partial^\mu \phi \partial_\mu \phi$
the latter notation is a bit sloppy, but widespread and standard and usually clear from the context. Also watch out for stuff like 
$ (F^{\mu \nu})^2 = F^{\mu \nu} F_{\mu \nu} $
