Fifth-placed Boston head to the CNG Stadium for both outfit’s third last game of the campaign.
Should Town get the better of their Midlands opponents, they will stretch out a seven-point advantage which would ensure them a top five place at the end of the season.
Weaver urged the home faithful to recreate the “cauldron” of support that roared Town towards four second half goals against Lowestoft Town earlier this month.
“It’s a great test for us,” he said.
“They have history of being up in the leagues above but that’s what we want. We want to make history and break new boundaries.
“It’s bound to be an exciting game as they are an exciting team to watch like us.
“Their fans are going to turn up in their numbers so it’s a great challenge for all of us.
“But we have that Kop and that pride in kicking towards it. We have to start the game well and get the fans reacting.”
Jack Emmett (ankle) is an injury worry while Andy Gascoigne and Dom Knowles are recovering from illness.
Tuesday night’s goalless draw at Nuneaton Town maintained a seven-point gap to their opponents, in sixth, and put Town on the verge of a highest finish under their current boss.
But the boss insisted he was sticking to mathematics and not getting ahead of it.
On whether Town had all-but secured their place with Tuesday’s draw, Weaver responded: “We are not quite in that position just yet but we are a step closer by not giving up any ground.
“There was a good buzz after. The lads really bounced back after a disappointing performance on Saturday.”
Town are also involved in the West Riding County Cup final on Tuesday night where they will face National League North rivals Bradford Park Avenue at Bradford City’s Valley Parade.
Weaver revealed he would entrust the young members of his squad to fight for the trophy as well as players requiring match fitness. Marcus Day, Lewis Riley and Rob Worrall are all likely to feature.
“It will give the younger lads a buzz to be playing at Valley Parade,” added Weaver.