Yes and it doesn't stay exclusive to OAs either. Any player that a Major Junior team moves off their active roster to their protected list can play Junior A (or B if they really wanted to) and can sign with another Major Junior team. There is no affiliations between Major Junior and Junior A teams, so when Major Junior teams say they reassigned a player, they are wrong.
For instance, I am a big fan of the AJHL's Lloydminster Bobcats. This season, the Prince Albert Raiders moved goaltender Nick Sanders to their protected list off their active roster, making him eligible to play Junior A or below in October. He signed with the Bobcats and played with them until the WHL trade deadline of January 10th, where Prince Albert traded his Major Junior rights to the Calgary Hitmen and the Hitmen decided to keep him on their active roster moving him from the Bobcats.
In fact, my Bobcats have had a run of problems in net this year. They started the year with Reece Klassen as their starting goaltender and in late September, he signed with the Lethbridge Hurricanes and has stayed as their backup since. They acquired Pierce Diamond from the Calgary Canucks and then Nick Sanders landed in their lap, which played out as I mentioned above. As they knew about the Nick Sanders deal before the deadline (the CJHL trade deadline is also January 10th), the Bobcats acquired Bailey Brkin's Junior A rights from the Brooks Bandits. He was playing with the Kootenay Ice, but became expendable when they acquired a goalie and was expected to be moved to the protected list in Kootenay. So he would have come to Lloydminster. However, Kootenay traded him to Spokane and they were still expecting Brkin to go to Junior A, the press release from Spokane ann0uncing the trade even said so. However, Spokane's starter Dustin Weatherill is injured and Brkin was kept in Spokane (and is playing well). Since the Bobcats cannot acquire another goaltender (and because of the way roster rules work in Junior A, they have to keep a card open for Brkin as he is expected to be moved to Spokane's protected list as soon as Weatherill is ready to go), they have to roll with Diamond and using Affiliated Players (i.e. Midget AAA typically) players as their backup in the meantime. The loss of Sanders really screwed up their plans as they were rolling along well just before he went to the Hitmen and were looking like they might be a darkhorse in the playoffs.
I like Diamond, he was a quality backup two seasons ago when the Cats went to the RBC Cup Finals behind Alex Leclerc. But he has been shaky as a starter this season and when you are stuck rolling through APs (who can only play in a maximum of 7 games before they need to be signed to a card) as your backup, there really isn't much the Cats can do. I suppose this is the case when Junior A teams rely on former Major Junior players instead of players who are looking at heading to the NCAA. If you are able to find a quality player who is intent on heading to the NCAA, it is generally a better option than a former Major Junior player, especially in net. However, it is a hit and miss situation sometimes as it means either going with the lesser talented option over someone who may end up being traded in Major Junior and vanish before the season is out.
Assuming Brkin ends up with the Bobcats, I am fairly certain he will stay. The question is if he will end up with the Bobcats at all this year. In all fairness to Junior A teams, Hockey Canada needs to move their trade deadline to a few days after the Major Junior leagues do their's. I know both the OHL and WHL had their set for the same day, the QMJHL seemed to have their's a few days earlier as no moves were made by teams in the Q on January 10th. This is the unfortunate part of things for Junior A teams because they are the highest level that Hockey Canada controls. The CHL is not under Hockey Canada control and thus has no need to work with them on anything except the agreement to allow players to join national junior teams. I have followed the Bobcats for a number of years and this is the first time I have noticed any of the AJHL teams lose three goaltenders in one season. So this is dumb luck that this has happened and the Bobcats need to muddle through. But it does suck that it happened to my team.
(Yes, I realize I answered your question and several you didn't ask lol.)