Boeing/MDD F-18E Super Hornet taxiing back after the Olympic Airshow 2012
boeing to my knowledge doesn't make fighter aircraft, they bid to make one that takes off vertically but lockheed nartin beat them with the f-22 raptor
The YF-22 (built by Lockheed-Martin) beat the Northrup-Grumman YF-23 in the Advanced Tactical Fighter competition back in 1991. The Boeing fighter you're thinking of is the XF-32, which lost the Joint-Strike Fighter competition to the Lockheed-Martin XF-35 Lightning II back in 2001.
the Hornet was originally a McDonnell Douglas design, but since Boeing bought them out, it became Boeing (just like the C-17, CH-47, etc)