Do harmonics work the same on bass guitar as is the case with a regular guitar (Natural harmonics at the 7th and 12th frets) or is it different on the bass?
It's the same as on a guitar. Harmonics occur at equal divisions of the string length. Half the string is the location of the 12th fret. This produces a harmonic at twice the frequency of the open string, which is one octave higher.
Dividing the string into thirds, which is at the 7th fret, produces the fifth of the 12th fret harmonic. (Halfway between the 7th fret and the bridge, at the 19th fret, you'll find the same harmonic.)
This continues to work up the harmonic series. If you divide the string in equal fourths (which occur at the 5th and 24th frets), you'll get a pitch two octaves higher than the open string.
By dividing a string, you can find these same harmonics on any stringed instrument, though you don't necessarily have frets as a convenient reference.
Harmonics are always at the same (proportional) spaces on any string.
The only thing that matters when determining where harmonics appear on a string would be length of the string. The 1st harmonic will always appear halfway inbetween the endpoints of the string (if it's fretted, count that as the endpoint), the next one divides the string into 3rds, etc...
That also means that the harmonics will always be in the same location relative to the fretboard (for instruments that have them), so as an example, the bass guitar and the guitar will both have a harmonic over the 12th fret on an open string.