What is the roman numeral assigned to an E♭ major chord in the key of F major? would it be a VII˚
It would just be ♭VII in most cases. The flat comes from you lowering the root of the 7th scale degree to make the chord and the upper case (i.e. VII instead of vii) is to signify that the chord is major. Depending on the exact progression, you could also be modulating. Typically these chords are viewed as borrowed from the Mixolydian mode.
X:1 L:1/1 M:none K:F V:2 clef=treble "bVII"[_E G B]|| %
Well, first of all the little circle after the VII means diminished, and a major chord is definitely not a diminished chord. An E chord in F major would be a viio (sorry I can't make the circle be above the line like it should be). There is no Eb chord of any kind within F major itself. So your Eb major chord is most likely a temporary modulation into another key, probably Eb or Ab.
Some analysts might say it was a IV of IV, that is, the IV chord of Bb major, which is in turn the IV chord of F. However, that is not a designation that most musicians would use or be comfortable with, so I think it should be considered a modulation.
Don't be surprised if someone else comes up with a different explanation though. My background is classical, and someone from pop, jazz, country, or rock might see things differently.
EDIT: Someone else posted an answer while I was writing this one, and sure enough had a different explanation. I'm betting they're from a different genre from me.
As people have pointed out, you would call that a ♭VII chord. The Roman numeral itself and the presence of an accidental determine the bottom (root) note of a triad. In this case, you take the seventh note in the F scale (E) and stick a flat on it (E♭).
This means that in the key of F, a ♭VII chord, a ♭vii chord, or anything else all start on E♭. The other stuff (capital letters vs. lowercase, diminished symbols, etc) determines what the notes above the root do. The diminished symbol (˚) means that you have a note a minor third above the root, then another note a minor third above that.
To give another example, a Db Major chord in the key of F would be a ♭VI chord.