Sounds like it could be a subconjunctival haemorrhage. Basically its where one of the small blood vessels in the sclera has burst. Generally not serious and it clears up in a couple of weeks.
I had one when I was working in Ipswich, the first I knew about it was when the one of the admin girls freaked out and dragged me off to one of the first aiders.
On balance it may have been wrong of me to look her in the eye at every opportunity while it was healing.
