A questions about negative resistance?
Okay, I take it you're identifying the 5V voltage source as the AA batteries you mention. Note that alkaline AA batteries produce 1.5V (nominal) each, so it could be 4.5V or 6V, but not 5V.

Also note that R4 does not appear in the Wikipedia circuit for a negative resistance converter, so you would need to do a network analysis to determine what the formulas would be and if indeed it still behaved like a negative resistance.

If you removed R4, (and perhaps if you didn't), current would indeed flow "backward", into the positive terminal of the voltage source as opposed to out of it. This would require of course that the V+ and V- supply to the op amp was high enough. What would be happening is the \$V_{out}\$ of the op-amp would be a higher voltage, say 10V, and current would flow from the output to the voltage source at the rate:

$$I_{rev} = \frac{V_{out} - 5V}{150\Omega}$$

Whether that current was enough to damage the batteries is a separate question; perhaps "How much reverse current can an alkaline AA battery withstand without damage?"

The op amp supply voltage will not affect the circuit behavior, unless it is insufficient to produce the required \$V_{out}\$. You'll have to do the math on that yourself, based on the particular op amp used and how close it can get to its rails. But if it works at say +/-10V, it would work the same at +/-12V or +/-15V.