The pKa of HBr is -9; the pKa of a protonated ether is approximately -3.6 (diethyl ether). Note pKa information can change by approximately 1 unit depending upon your source.http://evans.harvard.edu/pdf/evans_pka_table.pdf
If you combine anhydrous HBr (hydrogen bromide, not hydrobromic acid) with dioxane, on which side of the equation will the equilibrium lay? To determine this, you simply need a complete equation wherein you identify acidic and basic components, and then perform simple calculations.
When you have a species present to "drain off" the equilibrium, then you can expect it to happen. A protonated ether is an excellent leaving group when exposed to bromide as a nucleophile.
While "wikipremed" is cute, please read this article from the Canadian Journal of Chemistry:http://www.nrcresearchpress.com/doi/pdf/10.1139/v51-090
Ethers are readily cleaved by strong acids such as HI and HBr under anhydrous conditions. Cleavage with HI is far more rapid than with HBr.
Also, consult Advanced Organic Chemistry by Jerry March: