True or False?: There are infinitely many continuous functions $f$ for which $\int_0^1f(x)(1-f(x))dx=\frac{1}{4}$

Your argument has missed the fact that the domain of $f$ is the entire real line, not only the interval $[0,1]$. You have correctly shown that the restriction of $f$ to $[0,1]$ must be $1/2$, but there are infinitely many continuous functions on $\mathbb{R}$ with this property.

Here's an example based off of @Mike_Hawk's answer. Let

$$f(x)=\left\{\begin{matrix} \frac{1}{2} && \text{ for }x\leq 1\\ \left(\frac{1}{2}-b\right)x+b && \text{ for }x>1 \end{matrix}\right.$$

It is clear that $f(x)$ is continuous on $\mathbb{R}\to\mathbb{R}$ but that

$$\int_0^1 f(x)(1-f(x))dx=\frac{1}{4}$$

Since $b$ is a free variable, there are infinitely many continuous functions which satisfy your condition.