There is not a square date code. To me with loop it appears to be a triangle (1926?). I see several octagon Lone Eagle watches after the June flight. I also see in the old dated ads here listed with the tonneau shape as “Lone Eagle” . I read the history of the challenge to aviators from Bulova which raises yet another question? Are the specs on the watches sent to aviators catalogued at Bulova? The case number ends with 27, along with the patent date but not the movement code. The watch winds and runs and keeps time accurately for approximately 10 hours within one minute. It became a tight wind after being wound 10 times. I did not push it further, although I repeated this a dozen times with the same results. The dust cover is yellowed but intact. It is apparent that at some point a sloppy attempt to open the case was at the hinged end with some damage apparent. My takeaway is the watch deserves a day at the spa. The movement may be from 1926 and the case from 1927. The movement number looks early ending in 005. I can’t assume it is between Jan and June of 27 yet maybe possible with the research parameters available to me. I thank you for your time and look forward to your response.
a Case serial number beginning with a 1 indicates a 1931 manufacture date.
Is this the 'APOLLO' ?
I believe the ad is 1932 https://mybulova.com/sites/default/files/uploaded_photos/user2196/Sept.%201932.jpg
I agree that this watch is a 1931 Bulova Apollo. Sorry but it is not part of the Lone Eagle series, nor is it from 1927.
The JAN 11 1927 date is the patent date for the dust-proof cover that you mention and was common for a number of years.
There are 2 dial and hand variants. The first has a radium dial and hands as shown below.

The second, like your watch has raised golden numbers and 'modern' style hands as shown in this example.

When I wind my watches from this era I too will typically wind about 10 times before becoming fully wound. There is no point trying to wind any further once a watch is fully wound, you may damage it if you force it. My recommendation if you plan on wearing the watch over several days is to give it a full wind once a day around the same time. I typically do this first thing in the morning when I put my watch on.
A 1931 Apollo. Both case and movement are from 1931.
1931 Bulova Apollo is a good match.
Thank you all for your prompt replies in identifying my Bulova watch. I did not think the case date was for the Pat on the dust tile. The idea of a dust tile certainly deserves it. Considering the age of this Apollo and many other old watches I see on this site still functioning accurately, it is a clear testament to the innovation and preservation concepts implemented by a forward thinking manufacturer of fine watches for the years ahead.
In reply to Thank you all for your… by msaOlin11
It's certainy always been one of Bulova's strengths, to put quality on an equal footing as quantity. This is why affer 150 years the company is still going strong.