I have a circuit that acts as a switch between 2 LiPo batteries. I tested the circuit with 3.7V batteries on a breadboard. But now can I implement this using 22V 10000mAh LiPos on a stripboard or is the current too much ? Should I get a custom PCB instead?
Yeah, you should be able to do that. It depends on the amount of current that your circuit uses (Ohm's Law!)
If you are really worried, you can always put solder on the power traces, that will increase the amount of current it can handle as they will essentially be thicker traces.