Given $$ y=\frac{c_1 +c_2x}{c_3+c_4x} $$
Is there any test using the values of $c$ to see if between $x\in[0,1]$ there is at least one point where $y\in[0,1]$ other than just evaluating the function on the interval?
Examples could include using the distance between the hyperbola points or shifting the hyperbola.
Yes. You can easily manipulate the above formula to get an expression of the form $$y=A+\frac{B}{x+C}$$ What does this expression tells you? You can compare with $y=1/x$ hyperbola. The $C$ value just changes the vertical asymptote. The $A$ value just changes the horizontal asymptote. $B$ is a stretching factor in the vertical direction. Note that if $B<0$, you flip the hyperbola upside-down.
What you need to do is to calculate $y$ when $x=0$ and $x=1$, and calculate $x$ when $y=0$ or $y=1$. If none of these four points verify your requirements, there are no other options.