As indicated in the comments, it's not a question of grammar (even though it is a question about punctuation), it's a question of style.
In your example, Jack Russell is represented by a symbol: J.R. or JR (either with or without a space), or even a picture of a terrier. In that case, it really doesn't matter what the symbol is.
There is more of a question about whether full-stops should be used in names or for other abbreviations:
- Mr. J. R. Ewing
- Mr J R Ewing
- John R. Ewing III
- John R Ewing III
Current British practice is to omit many of these full stops. It's fussy and they aren't really necessary to show that the word is an abbreviation: it's unlikely that Mr Ewing's middle initial is just "R". I generally use my middle initial, and never use a full stop. Even a less obvious abbreviation like viz is (it appears, subjectively) losing its customary full-stop.
I suspect the reason for this is that additional stops
- need interpreting to see if they are the end of a sentence
- take an additional keystroke, or time to write
and both of those are reasons in this hectic age to omit dots.
But this is still a matter of style: to use a full-stop may be prescriptively required, but it does nothing to enhance the descriptive function of an initial or other abbreviation.