Can homeowners attend closed executive board meetings?

Yes they can but it is up to the board.  Here is what Arizona’s Open Meeting Laws say:

33-1804. Open meetings; exceptions

A. Notwithstanding any provision in the declaration, bylaws or other documents to the contrary, all meetings of the association and board of directors are open to all members of the association or any person designated by a member in writing as the member’s representative and all members or designated representatives so desiring shall be permitted to attend and speak at an appropriate time during the deliberations and proceedings. The board may place reasonable time restrictions on those persons speaking during the meeting but shall permit a member or member’s designated representative to speak before the board takes formal action on an item under discussion in addition to any other opportunities to speak. The board shall provide for a reasonable number of persons to speak on each side of an issue. Any portion of a meeting may [NOT MUST] be closed only if that closed portion of the meeting is limited to consideration of one or more of the following:

1. Legal advice from an attorney for the board or the association. On final resolution of any matter for which the board received legal advice or that concerned pending or contemplated litigation, the board may disclose information about that matter in an open meeting except for matters that are required to remain confidential by the terms of a settlement agreement or judgment.

2. Pending or contemplated litigation.

3. Personal, health or financial information about an individual member of the association, an individual employee of the association or an individual employee of a contractor for the association, including records of the association directly related to the personal, health or financial information about an individual member of the association, an individual employee of the association or an individual employee of a contractor for the association.

4. Matters relating to the job performance of, compensation of, health records of or specific complaints against an individual employee of the association or an individual employee of a contractor of the association who works under the direction of the association.