Is there any way I could use the native log backups that it generates with SQL Backup log shipping job running on the destination server?
Theoretically you could modify the jobs created by SQL Backup logshipping to use native SQL Server syntax for backup and restore.(I wouldn't recommend this.)
In other words, can the Restore command handle a native backup?
The SQL Backup restore command will not be able to handle the native backup file.
Is there a converter program from native to .sqb?
We do not have a converter for .bak to .sqb.
I would recommend that you use native log shipping in your case.
Here is an article I found that should point you in the right direction to get that set up.
http://www.mssqltips.com/sqlservertip/2 ... -shipping/