The Barkhausen criteria are usually applied to analyze sine wave type oscillator circuits (Wien bridge, etc.) where a small signal, such as thermal noise, is exponentially amplified around the positive feedback loop to create the output signal. Oscillation is inherently a large signal phenomena and in general can't be analyzed using LTI analysis methods, but the Barkhausen criteria let you predict oscillation from the small signal gain and phase behavior.
It's less clear to me how to directly apply such techniques to this relaxation oscillator circuit, as circuits like this don't have any small signal behavior - there are only 2 stable states. It should be fairly obvious, however, that whatever component values you choose the feedback around the loop will eventually be unity and in phase, i.e. when V+ = V-.
It is a long question, but better than a short one, as you've shown your own research.
1) Solar cells. If you're stacking your own ones, stack 9 of them and get the 4.5V of the original circuit.
2) Battery charging. Batteries are the only thing you've left out of your spec. This is an area where the circuit design relies on cutting a lot of corners. In theory it might be out of spec, if you were to put 4.5V at 280ma through AA NiMH cells indefinitely. In practice, you don't get full sun all day, you'll be using it indoors, and you're not going to get optimal power transfer from the cells, so this isn't going to cause problems.
3) Diode. It's just a regular diode, not a zener. Current through it is actually determined by the battery and right hand side circuit, not the solar panel - the transistor is off when the panel is generating electricity. The original 1N914 will be fine. 1N4004 will also be fine.
4) Resistors: not a precision component here, use whatever meets your cost constraint. 5.1k for 5k is fine.
5) Wire: not critical. Your ebay link looks suitable. Thinner is better for the toroid.
6) Transistors: stick with the exact part numbers. Design may rely on specific parameters.
7) LED: again, this circuit relies on cheating. Normally a white LED won't run from two NiMH cells. The joule thief part provides a boost converter that gives small pulses of higher voltage. It doesn't have the capacity to provide a lot of current at that voltage. In combination with the pulsing this means there should be no risk of damaging it.
(A proper analysis of this circuit would be good, if nobody else supplies one I'll do it in a few days).
Best Answer
Yes it should work, assuming the sensors are well matched. Allow for some leeway. Eg require the signal to reach a minimum difference between the sensors before outputting HIGH. If you give more info, eg purpose of system, what's driving /controlling the laser, and that your final output is used for, I could help with a circuit.
You may be able to avoid needing two sensors if you encode the laser, or at least pulse it. For example, have it pulse at 1kHz (assuming your sensors react fast enough). Then filter out signals below that, then smooth that result and add a schmitt trigger for your final logic signal. This is the idea behind how IR remote control receivers block out ambient light from interfering. The underlying frequency, eg 1kHz is called the carrier frequency.
Either way, using a light filter which blocks everything but the laser light colour could help.
Edit: I found this circuit from here which looks near perfect:
As per my comment, just ignore the components after the final op-amp and feed into the MCU instead. Then just change a few values to change the reception frequency (currently 40kHz).