I currently have a very old White Rodgers thermostat with large push buttons at the very top.

I want to replace it with a Hunter Model 44260.

Here is the current configuration
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red <= Don't know if RED goes to RC or RH
white
green
yellow
black <= Don't know what to do with the BLACK wire

I have a separate furnace and air conditioner.

Hope you can help. Thanks!

Are you talking about heat and a/c only ,or is it a heat pump?, the rule of thumb is red to rh, white to w,green to g,yellow to y, the black well,, I would look inside the furnace and see what or where the other end of the stat wires are connected to,, see if the black is even being used.

Since you said you have a furnace, I see no need to worry about heat pump. If you only have the usual one transformer, the red goes to both RH and RC. Likely the black is a common which most thermostats don't need. If the new Hunter does, make sure it is connected to the com terminal at the furnace. It is connected to the other side of the transformer from the red and provides a return path for all th circuits. The others you have right.