I'll be at my uncle's house in case you need to reach me.
I'll be at my uncle's house just in case you need to reach me.
I'll be at my uncle's house should you need to reach me.I'll be at my uncle's house you should need to reach me.
Could you tell me what the difference between these is?
In addition, when can I invert "should" in such a sentence?
UPDATED: I am wondering what is the exact difference between the bold parts?
∗ I'll be at my uncle's house you should need to reach me
is ungrammatical. You need an if in there: if you should need to reach me. The if is only omitted if should is inverted with the subject. The inverted construction is a ‘fossil’ from much older usage; today it is fading into disuse, and has a very quaint ring in the US. Once any verb might be used in this construction, but today only should, had and were are used this way.
The rest of your sentences are all ‘relevance’ conditionals: that is, the protasis (‘condition clause’, ‘if clause’) does not express a condition under which the apodosis (‘consequence clause’, ‘then clause’) is true but a condition under which the information in the apodosis is relevant. For the most part they all express the same thing, which may also be expressed with an ordinary if clause:
I’ll be at my uncle’s house if you (should) need to reach me.
However, there is another dimension to the versions with (just) in case. In case may express, alternatively or additionally, the reason why you will be at your uncle’s house: you will be remaining at a particular place (rather than going out) so that you can be reached if that becomes necessary. That is the primary meaning of the expression, as in
I’ll take an umbrella in case it rains.
I'll be at my uncle's house just in case you need to reach me.
The person who said this sentence informed the other one that "if you ran into any kind of trouble you can contact me in my uncle's house because you'll find me there."
I'll be at my uncle's house should you need to reach me.
Formal sentence; meaning that "if you needed to reach me, know that I'm at my uncle's house."
I'll be at my uncle's house you should need to reach me.
Odd sentence :) , meaning that "I suggest you reach me, I will be at my uncle's house."
Notice that these three statements follow very different grammatical structure. The first one is two statements linked with "just in case" conjunction, the second one is an inverted conditional, and the third one, well, it seems to me that it's two separate statements combined with no conjunctions!
Update: The use of "just" makes no vast difference in the meaning of the sentence, it can be ignored. Hey, here's a helpful page about inversions, were you eager to learn about them. Hope I've helped.
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.
