Can you most always shorten the word "oubyen" to "ou", following the same rule as in the word "genyen" to "gen"? I have not heard of "ou" being used as much as "gen". Mwen pa konprann. Mesi anpil.

Actually "ou" is not a shortened form of "oubyen" .
In H. Creole, we say ou (from French "ou"), o, oubyen (from French ou bien), onswa, oswa or ouswa (from French soit) and osnon or osinon (from French ou sinon) all which basically mean "or", "either", "neither", or "rather"

We do use all these translations for "OR" quite often.
You might hear them in sentences such as:

1. Mwen bezwen youn ou de chèz.
    I need on or two chairs.

2. Kilè w'ap tounen?  Novanm ou desanm?
    When will you be back?  November or December?

3. Kilès ou vle?  Ble a oswa wouj la?
    Which one do you want?  The blue or the red?

4.  Ban m onswa kafe ou te.
     Give me either coffee or tea.

5.  Ou kapab sèvi  oubyen Bondye ou Satan. Men ou pa ka sèvi toulede.
     You can either serve God or the devil.  You can't serve both.

6.  Onswa ou renmen'm oubyen ou pa renmen'm.  Li pa ka toulede.
     Either you love me or you don't.  It can't be both.

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