Yes, you can definitely develop a sore throat
from kissing someone. Most sore throats (or pharyngitis) are caused by viral pathogens which are most commonly passed from one person to another through close contact, or the passing of bodily fluids (like kissing).
This does not have to be the case, however, as there are definite cases where bodily fluids containing the virus end up on hands, etc (for example after a sneeze) and then due to lack of good hand washing techniques the virus can be spread from person to person. This is why currently in most medical hospitals there has been a large hand washing initiative to help prevent the spread of infection. There is something called the prodromal phase of viral replication which is the in vivo (meaning replication within a person) replication of the virus without the full spectrum of the active symptoms caused by the virus.
In other words it is the time between getting infected with the virus until you are actually sick from it. This may explain the 2 day lag between feeling sick. If your boyfriend truly did have influenza, I would recommend getting evaluated by your primary care physician
. Hope this is informative.