Reaction of 1-(1-chloroethyl)-1-methylcyclobutane under E1 Conditions
You are correct that an alkyl shift will take place but there are two possible options for which group moves. Your option is possible but there is another option which involves a ring expansion to generate product A.

This mechanism is favoured because it relieves the ring strain in the four membered ring. Since the reaction is run under reversible conditions, the thermodynamic product (with the least ring strain) will dominate.