The last name of Elizabeth (Vruggink) was a great help!
In the Antwerp Police Immigration Index, 1840-1930 (to be found on familysearch) I found the following persons: Elisabeth Hubertine Vruggink Bernard Henri Apol
Searching through various Dutch databases I found the following: Elisabert Hubertina Vruggink was born in Roermond on 10 Oct 1855, Daughter of Hendrikus Vruggink and Johanna Haekens. Bernardus Henri Apol was born in Dordrecht on 25 Aug 1848.
These dates match the birth dates of “your” Henry Apol and Elizabeth Apol in various US documents, so I am reasonably certain (not 100% though) that Bernardus Henri is the same person as the Henry you are looking for.
And now for the family connection between Louis and Bernardus Henri: After a lot of searching I found they were cousins, their fathers were brothers.
The family of Bernardus Henri Apol:
Cornelis Apol, copper and tin smith / manufacturer / lamp maker Born Breda 08 Jan 1807, bapt. 11 Jan 1807, died ? Son of ADRIAAN APOL and ADRIANA CHRISTINA HARMSEN Married Breda 05 June 1840 to: Joanna Henriette van Ketel Born Westbroek (province of Utrecht) 13 Oct 1813, died Dordrecht 03 Apr 1850 Daughter of Jan Hendrik van Ketel and Geertruida van der Vangst Children: 1. Cornelis Hendrikus Apol, born Breda 27 Feb 1843 at 12.00 hrs 2. Johannes Gerardus Apol, born Dordrecht 18 Aug 1845 at 02.30 hrs, died 27 Aug 1845 3. Bernardus Henri Apol, born Dordrecht 25 Aug 1848 at 18.30 hrs
After the death of Joanna Henriette, Cornelis remarried in ‘s Gravenhage (The Hague) on 15 Aug 1860 to: Aagje van Rooijen (seamstress), daughter of Elias Willem van Rooijen and Cornelia Elisabeth Colet
The family of Lodewijk Franciscus Hendrik (Louis) Apol:
Johannes Cornelis Apol, art painter Bapt. Breda 03 Feb 1806, died Velsen 04 Mar 1896 Son of ADRIANUS APOL, shopkeeper, and ADRIANA CHRISTINA HARMSEN Married The Hague 31 Mar 1847 to: Antonia Johanna Hage Born Amsterdam 07 June 1821, died Heemstede 05 Nov 1903 Daughter of Hendrik Hage, book printer, and Johanna Paulina Prinses Children: 1. Anna Maria Apol, born 22 Oct 1845 at 11.00 hrs Born as A. M. Hage, acknowledged by father at marriage parents, therefore after that with last name Apol. 2. Johannes Adrianus Cornelis Apol, 24 Oct 1848 at 03.00 hrs 3. Lodewijk Franciscus Hendrik Apol (Louis), born 06 Sep 1850 at 10.00 hrs
I didn’t find the marriage of Bernardus Henri Apol and Elisabeth Hubertina Vruggink, though. But they may have gotten married in Antwerp….
I suggest you post a message on the Belgium board re. this marriage and the births of the children Annie (June 1878), Bertha (Aug 1879), Rosa (Aug 1882) and Arthur (Jan 1885). The population registration (in Dutch bevolkingsregister) can be helpful as well. In there it is stated what their last residence was, what the date of their marriage is (and if you are lucky where they were married), the names and dob of their children, etc. If you get all that, you can really be sure that Bernardus Henri is indeed Henry!