Is the experimental evidence confirming black hole entropy or Unruh radiation? The question says it all: how does Bekenstein–Hawking entropy or radiation fare when compared with observation?
Or maybe just the idea of Unruh radiation?
 A: There is no experimental evidence for or against Hawking radiation. For realistic black holes, the effect is far too small to measure with current or foreseeable technology.
There is also no universally accepted evidence for the Unruh effect. It can also be hard to pin down exactly what constitutes a test of the Unruh effect (see Matt Visser's article in https://arxiv.org/abs/gr-qc/0102044). Directly testing it by accelerating a macroscopic observer is impractical.
A: When it comes to direct observation of actual black holes, I agree with Andrew's answer. In this answer, I'll address a different possibility: analogue models and consistency checks with other well-tested theories. Notice these are not direct experimental evidence, but they can still be interpreted as experimental evidence.
Hawking Radiation: Analogue Black Holes
While it is incredibly difficult to experimentally measure Hawking radiation in an actual black hole, one alternative possibility is to consider analogue models. These are physical models that do not correspond to gravitational physics problems, but can be written as if they did. Let me give some examples.
Suppose you have an ideal fluid flowing in a spherically symmetric manner inwards to the origin. One can show (Unruh 1986; Barceló et al. 2011) that the equations of motion for linear perturbations of such a fluid (i.e., sound waves) under some conditions respect the very same equations a scalar field would respect in a black hole–like spacetime. In this model, the analogue black hole is a sonic black hole: there is a region where the fluid flow is faster than the speed of sound on the fluid, meaning sound waves get trapped and can't escape that region (pretty much like nothing can escape a black hole). By quantizing these perturbations and proceeding as usually done with Quantum Field Theory in Curved Spacetimes, one gets a prediction of Hawking radiation in this analogue situation.
Now, this analogue radiation has recently been observed in condensed matter systems (Muñoz de Nova et al. 2019). Of course, this is not a direct observation of the actual Hawking effect in black holes.
Some References:

*

*Unruh, W. G. (1986) Experimental Black Hole Evaporation? Physical Review Letters 46, 1351–1353. DOI: 10.1103/PhysRevLett.46.1351

*Barceló, C., Liberati, S. & Visser, M. (2011) Analogue Gravity. Living Reviews in Relativity 14, 3. DOI: 10.12942/lrr-2011-3.

*Muñoz de Nova, J.R. et al. (2019) Observation of thermal Hawking radiation and its temperature in an analogue black hole. Nature 569, 688–691. DOI: 10.1038/s41586-019-1241-0
The Unruh Effect is Mandatory: The Proton's and Larmor's Testimonies
The title of this section is a reference to Matsas & Vanzella 2002. While there is no direct experimental evidence of the Unruh effect, it is mandatory if you are not willing to challenge Quantum Field Theory in flat spacetime and Classical Electromagnetism.
Matsas & Vanzella 2002 consider the case of the decay of an accelerated proton in both an inertial reference frame (which uses only standard QFT, common to Particle Physics) and in the proton's reference frame, which is accelerated. Consistency of the results requires the Unruh effect to be present in the proton's reference frame.
Similarly, Cozzella et al. 2017 considers an accelerated charge. In classical electromagnetism, an accelerated charge emits radiation. By considering how a charge interacts with a thermal bath in the charge's reference frame and comparing with the predictions of classical electromagnetism made in an inertial reference frame, one arrives at the conclusion that the Unruh effect is necessary to keep consistency of both descriptions. Quoting the paper's abstract,

Here, we propose a simple experiment reachable under present technology whose result may be directly interpreted in terms of the Unruh thermal bath. Then, instead of waiting for experimentalists to perform the experiment, we use standard classical electrodynamics to anticipate its output and show that it reveals the presence of a thermal bath with temperature TU in the accelerated frame. Unless one is willing to question the validity of classical electrodynamics, this must be seen as a virtual observation of the Unruh effect. Regardless of doubts still raised by some voices, the Unruh effect lives among us.

Some References:

*

*Matsas, G. E. A. & Vanzella, D. A. T. (2002) The Fulling–Davies–Unruh Effect is Mandatory: The Proton's Testimony. International Journal of Modern Physics D11, 1573-1578. DOI: 10.1142/S0218271802002918. arXiv: gr-qc/0205078.

*Cozzella, G., Landulfo, A. G. S., Matsas, G. E. A. & Vanzella, D. A. T. (2017) Proposal for observing the Unruh effect with classical electrodynamics. Physical Review Letters 118, 161102. DOI: 10.1103/PhysRevLett.118.161102. arXiv: 1701.03446 [gr-qc].

