I'll be at my uncle's house in case you need to reach me.
In case means “If the following possibility occurs.” A “case” in this sense is one of a set of possibilities considered during planning, or one of a set of possible outcomes.
I'll be at my uncle's house just in case you need to reach me.(colloquial)
In this context, the word just functions as a “softener”. Just in case suggests that the following possibility, while bad, is unlikely. The central meaning of “just” (in this sense) is like “only” or “a little bit”. Combined with in case, it means that the possibility is a small thing, nothing to worry about, but you’re still addressing it.
I'll be at my uncle's house should you need to reach me.
I explained the effect of should in detail in this answer “just” a few minutes ago. In this context, should introduces a hypothetical condition where the the situation has veered away from happy/normal/good in some way. If you included if (“…if you should need…”), should would soften the hypothesis by suggesting that the you’re unlikely to need to reach me or by suggesting that you need not worry, because even if you need to reach me, we are ready to solve the problem right away.
I'll be at my uncle's house you should need to reach me.
This one is ungrammatical. When you use should in place of if, you need to invert the subject and auxiliary verb as you do when forming a question.