Does it make sense to store a couple of Boolean values as array?

TL;DR: Do not use an array. Use individual boolean columns.

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Your coworker may not be aware of actual storage requirements. Null storage is very cheap and efficient in Postgres.

• Do nullable columns occupy additional space in PostgreSQL?

A couple of boolean columns are also very cheap and efficient - nullable or not. Actually, just 5 boolean columns hardly matter for the row size at all - even if defined NOT NULL. 1 byte each, 5 bytes and no alignment restrictions. Hardly relevant. There are typically much more substantial things you can do to your row. (You should have provided your actual table definition.)

• Configuring PostgreSQL for read performance

Arrays have an overhead of 24 bytes. Plus actual data. And handling is much less convenient and much more expensive:

• Calculating and saving space in PostgreSQL

This related answer has a detailed assessment for three relevant options:

1. separate boolean columns
2. an integer to encode up to 32 boolean values
3. a bitstring (bit(n) or bit varying(n))

• Is there any difference between integer and bit(n) data types for a bitmask?

In your case I think you could consider the use of a Bit String Type data type. For instance, something like:

CREATE TABLE yourtable (
   booleans bit[5] default B'00000',
   ... other fields ... )

It is efficient in terms of memory and does not require the use of a complex type like a PostgreSQL array (actually it is a bit array), and more, you do not have to pay attention to the difference between false and null (and also you could set the entire field to a null value, if you need to).