[Gabriel Labelle]"What can I do to preserve the quality of the still image?"
Change you sequence settings to ProRes422. H264 is not an editing codec and although FCP lets you set it as a sequence codec, you just shouldn't. Nor should H264 footage be used in the timeline. H264 is highly compressed and won't give you proper playback. It may also cause FCP to crash. Apart from that it will crunch further a jpeg still image.
All stills should be set to 72dpi btw. Walter Biscardi wrote a tutorial recently explaining why if you want to do a quick search. While you are searching, also look at H264 and you will see what how popular that codec is with FCP users and how you should ingest material into ProRes before editing.