Can you use both sides of an equation to prove equality?
There's no conflict between your high school teacher's advice
To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side.
and your professor's
To prove a statement is true, you must not use what you are trying to prove.
As in Siddarth Venu's answer, if you prove $a = c$ and $b = c$ ("working from both sides"), then $a = c = b$ by transitivity of equality. This conforms to both your teacher's and professor's advice.
Both your high school teacher and university professor are steering you away from "two-column proofs" of the type: \begin{align*} -1 &= 1 &&\text{To be shown;} \\ (-1)^{2} &= (1)^{2} && \text{Square both sides;} \\ 1 &= 1 && \text{True statement. Therefore $-1 = 1$.} \end{align*} Here, you assume what you want to prove, deduce a true statement, and assert that the original assumption was true. This is bad logic for at least two glaring reasons:
If you assume $-1 = 1$, there's no need to prove $-1 = 1$.
Logically, if $P$ denotes the statement "$-1 = 1$" and $Q$ denotes "$1 = 1$", the preceding argument shows "$P$ implies $Q$ and $Q$ is true", which does not eliminate the possibility "$P$ is false".
What you can do logically is start ("provisionally", on scratch paper) with the statement $P$ you're trying to prove and perform logically reversible operations on both sides until you reach a true statement $Q$. A proof can then be constructed by starting from $Q$ and working backward until you reach $P$. Often times, the backward argument can be formulated as a sequence of equalities, conforming to your teacher's advice. (Note that in the initial phase of seeking a proof, you aren't bound by anything: You can make inspired guesses, additional assumptions, and the like. Only when you write up a final proof must you be careful to assume no more than is given, and to make logically-valid deductions.)
It is enough.. Consider this example:
To prove: $a=b$
Proof: $$a=c$$ $$b=c$$ Since $a$ and $b$ are equal to the same thing, $a=b$.
That is the exact technique you are using and it sure can be used.
short answer: equality is symmetric, implication is not (both are however transitive)
longer answer:
You are right: if you proof A = C and B = C for some terms/expressions/objects A,B,C then you are allowed to conclude that A = B (because "=" is transitive and symmetric)
Your teacher is right: if you prove that something true follows from A = B, i.e. A = B => true, you are not allowed to conclude the converse i.e. that A = B since "true is true" (because implication is not symmetric)