We show that the square of every connected $S(K_{1,4})$-free graph satisfying a matching condition has a $2$-connected spanning subgraph of maximum degree at most~$3$. Furthermore, we characterise trees whose square has a $2$-connected spanning subgraph of maximum degree at most~$k$. This generalises the results on $S(K_{1,3})$-free graphs of Henry and Vogler (1985) and Harary and Schwenk (1971), respectively.