That Is a tough one. Generally speaking, " I can't go on" would be an expression to say something like "I can't take any more", basically saying they are burned out, giving up. However, to be a band with a name " I can't go on, I'll go on" seems like they are playing around with two expressions; " I can't go on" / " I must go on". " I must go on" would be, of course, a follow up to someone expressing they are tired or exhausted but then says " I must go on". Or you might hear "The show must go on" an expression stating that giving up is not an option.

I'm not sure why the band is named that. Looking at the sentence by itself, it may not make much sense. The reader has to think about what it can mean. This is my idea: "I feel like I can't move forward with my life (because things get difficult). But still, I have to try."