Short answer is No.
Part of the SQL installation must go on the system partition.
If your goal is to make SQL portable with the drive, then that is not going to happen. I would install SQL locally to run on the laptop. I would then change the SQL Data and SQL Log file locations to the external drive. If SQL Starts up
and the disk is not installed, the databases will show up as SUSPECT. Attaching the external HD and restarting SQL will bring the databases back to normal operation. Make sure and shut down SQL Server before detaching the external drive.
Geoff N. Hiten Principal Consultant Microsoft SQL Server MVP