Differential entropy can be negative for some functions, but when the domain is an interval of length $1$, I suspect it can not be.
Claim. For any probability density function $f$ with domain $(0,1)$,
$$-\int_0^1 f(x) \log f(x)\ dx \geq 0.$$
Is this claim true?
Note. I omitted a minus sign everywhere and the question became misleading. I did not edit this question because that would have affected the accepted answer. Instead, based on this meta recommendation, I wrote this clean version for better reference.
Nope. The uniform distribution $U(0,a)$ for any $0<a<1$ has differential entropy $\log(a)$, which is negative.
If you want something that is supported on exactly $(0,1)$, you can consider the average of two independent uniforms on $(0,1)$, which follows a triangular distribution: $$f(x) = \begin{cases} 4x & x \in (0, 1/2] \\ 4(1-x) & x \in (1/2, 1).\end{cases}$$ The differential entropy is $1/2 + \log(1/2) \approx -0.19.$