I have a old rheem furnace. I recently went to see why it hasn't been working properly and noticed that my dogs had gotten into a junction box on it that had wires from the fan motor, fan and limit control and power switch all running to it. Each had 2 wires (black and white) Should I just connect the 3 black wires together and then the 3 white wires together? Or am I asking for troubled?

Without seeing a wire diagram you can't be sure but what you are considering doing is the normal pattern. Make sure power is off. Was the jbox covered?