The easiest way, and most likely to succeed, is to pay when you check in at Logan airport. This will depend on (a) there being space in PE or UC cabins, and (b) paying a large sum of money.
Free upgrades for a group of four or more are incredibly rare, especially from the economy cabin. Increasingly airlines only offer upgrades when they have to make seats available for an overbooked cabin. For example, 32 seats in Premium Economy and 33 tickets sold on the basis that you will normally get at least one no-show. However, if all 33 people turn up one will have to be upgraded or offered incentives to move to a later flight. It follows that the smaller your group (ideally, just one person) will give you the best chance of being upgraded.
Airlines have generally stopped offering free upgrades just because there are seats available in business or first. There are some exceptions to this. Virgin, for example, offer free upgrades to honeymooners (if you tell them in advance?) and some people who are classed as CIPs (commercially important passengers) because they work for a company which has a huge travel bill (I worked for a company like this and was upgraded a few times).
Sorry not to be more hopeful, but I think the chances of an entire family getting upgraded free of charge from Economy is slim.
Regards
Rob