If you found more than 5 queen cells, that usually means the bees swarmed. No eggs definitely means no queen! If you found fewer, then the bees may have superceded this queen. Happens even with a new package.
2 things to do: I like to leave them only a couple of queen cells, and either take out the others into another hive (or in my top bar, I would put on the other side of the hive with a bee-tight divider in place), or I would just squish more beyond my target number. Bees will throw off another swarm if they have a virgin queen emerge, AND there are more than 10 Langstroth frames densely covered with bees. Of course, some hives don't do a secondary swarm, and others will keep throwing swarms as long as there are bees in the hive - but for a typical case, a hive that swarmed but still has lots of bees in 5-7 days will often throw a secondary swarm headed by a virgin queen. You don't want that and you can stop it from happening.
Second, you need to draw up a calendar of when the queen is likely to be back from being mated. So, I saw from the pic you had a capped queen cell - it was maybe at day 10 when you posted, on 6/12. By day 10, I mean 10 days after the queen laid this egg. Since the queen cell is still pale, it is early on its journey to ripen and discharge a virgin queen.
On day 16, queen is out (so, 6/18). About 5 days later, she goes for her mating flight. Give her 5 more days after that, for her to recover and reconfigure her body to lay eggs...so at the earliest, 6/28, she will start laying. By 7/2 you should see eggs for sure. Assuming you had clear weather during her flight day(s); if not, things will be pushed back. During the likely flight time, I take notes on whether the weather would permit a mating flight - has to be 70, fairly calm, no threat of rain, from 1-4 pm or somewhere in there.
So by 7/2, if no queen is laying, you have maybe 10 days before that hive starts being a laying worker hive. Then it is not possible to requeen, because they will just kill an introduced queen. You can save such a hive with brood introduced every 7 days, about 3 times. AFter that process, you could introduce a queen.
So you want to avoid that - you need to have a plan for getting a queen or queen cell quickly by 7/5 or so. Your local bee association and local beeks will know of queen sources, or if someone can graft, they can bring you an artificial queen cell. But it's better to start the legwork - I don't mean actually ordering a queen, but finding out where you can get a queen a short notice, or a queen cell - at this point rather than later.
Good luck! Hope she makes it back ok!