I'm not an expert, but I would create a calculation field, using Cases, with each case defined by which of the boxes are "empty" or "not empty", returning text results that describe what you are looking for. eg "One and Five only". Be sure to select TEXT as the Calculation result type.
Then make a value list of these returned values and assign it to the field in Field Format. Then search on that field.
I do this a lot for date or progress status finds, based on which date fields have values (empty or not empty).
Enter find mode
Click the first check box
Select "New Request" from the "Requests" menu
Click the third check box
Perform the find.
Entering criteria in different requests like this sets up your OR search. This can also be scripted. See the looping script near the end of this thread: Scripted Find Examples
Extend found set can also be used.
Yes, thanks! I tried extend found set method and it seems to be working, however now it searches only by check boxes fields. And ignores parameters for search from the others fields. Can you please check my script?
I'm not too much familiar with coding, so I just little edited and inserted in my script the piece of code from Scripted Find Examples . Now it looks like this:
Set Error Capture [On]
Perform Find 
Set Variable [$rooms; Value:household:room_number]
Enter Find Mode 
Set Variable [$K; Value 1]
Set Field [household::room_number; GetValue ( $rooms ; 1)]
Set Variable [$K; Value: $K + 1]
Exit Loop If [$K > ValueCount ( $rooms ; $K )]
Extend Found Set 
If [Get (LastError) > 0]
Modify Last Find
Show Custom Dialogue ["Error"; "No records found."]
Exit Script 
New Window 
Go to Layout ["household" (household)]
... next go new window parameters
Where "household" is my db, and "room_number" is my checkbox field
To combine "and" criteria with "or" criteria, specify the "and" criteria each time you extend the found set along with the "or" criteria that is different each time.
And keep in mind that extend Found set adds the records found to those already in the found set so your first find has to use Perform FInd instead of extend found set or you use other steps to create a found set of zero records before you start extending the found set.