No.
The R-SH group is nucleophilic. Br- is a good leaving group. If you have a good nucleophile and a center with a good leaving group present, what can happen?
For example, if you had R-CH2OTf (-OTf is an excellent leaving group) and you threw in some NaI (I- is an excellent nucleophile) you should expect the nucleophilic I- to attack R-CH2OTf in an Sn2 fashion and form R-CH2I.
The blue compound you drew after eliminating HBr is wrong, it would be a + charge left behind, ie a secondary carbocation, not anion - but I think you should be mainly looking at other substitution mechanisms... nudge nudge wink wink...