I checked the Brålanda birth records for you to try to find the birth of an August there in November 1866. (They are online on the fee-based Genline site.) I don't see his birth.
The only August born in November 1866 in Brålanda parish was the son of Peter Jonsson. If August had used the "modern" trend of having an actual surname, he would have grown up to be August Jonsson and if he had used the normal custom of having a patronymic name instead of a surname, he would have grown up to be August Petersson (son of Peter). He can't be yours.
I did find a Johan born on 24 August 1860 who was the son of Anders Johansson and Cajsa Magnusdotter. That Johan could have grown up to be Johan Andersson (son of Anders). Maybe he is yours. (The A in John A. Bergquist would have most likely been the remnant of his patronymic name of Andersson.) This is where you could see that birth record if you joined Genline.
I'm surprised that there wasn't a specific birthdate in the church records of that Lutheran church you mentioned. Lutheran church records were ***extremely*** detailed in those days since the ministers came from Sweden where they had been trained in the rather nitpicky ways of the official Swedish church. I've even seen the year of leaving Sweden listed in their records and the date of joining and leaving the church should be listed too.
I realize that you are more interested in the U.S. part of your message but the parish Brålanda caught my eye. I have deep roots going back many generations in Brålanda parish and a nearby parish (Frändefors) and shallow roots in Gestad, another parish nearby. There is always the potential of finding distant relatives so I like to check, just in case. I don't think this family fits my tree but we never seem to end our research, so I still like to check.
At any rate, the Johan found above did emigrate from Sweden. Maybe he is yours. I found him on the Emibas CD. Emibas was made from registrations in the Swedish parish to go to a foreign country. It is about 75% complete.
Andersson, Johan Man
b. 8/24/1860 in Brålanda, Älvsborgs län (Dalsland)
Emigrated 2/10/1882 from Låttsbyn, Brålanda, Älvsborgs län (Dalsland) to Amerika
Source: Emibas migration file ID: Brålanda P 1882 003
I checked a different parish record called the husförhörslängd (HFL). The HFL is usually translated as clerical survey record or household examination records. It has the format of a census but it was far more detailed and was updated continually instead of once every ten years. I checked the HFL for 1876-1880 to try to find August together with your Johan. I do see a son named August born in November but the year was 1863 and not 1866.
GID # 586.39.65900
lodger Anders Johansson, born 30 September 1832 in Brålanda
wife Anna Cajsa Magnusdotter, born 18 January 1832 in Steneby
Johan, born 24 August 1860 in Brålanda
August, born 4 November 1863 in Brålanda
Augusta, born4 October 1856 in Brålanda
Try to find specific birthdates for your John and August so that we can compare with the above or else find other possibilities for you.