Uiua

Not really proud of this one. Part 1's still ok, just calculated all distances between boxes after realizing it's not that many (499500, a reasonable amount I think, relatively speaking).
The dumbest thing I did this time was manually implementing the function to take the first n rows of an array by using a loop. Only when I was working on part 2 did I realize that I can just use the "take" function Uiua provides. Additionally, I even had some mistake in my reimplementation of it that only caused issues in part 2 somehow.
For anyone interested, it's the commented out line in P₁ below.
Part 2 is just a brute force. For the actual input, I searched manually until I got to the pair number just before the actual solution because I didn't want it to run that long and it takes long enough as is.

Run with example input

$ 162,817,812
$ 57,618,57
$ 906,360,560
$ 592,479,940
$ 352,342,300
$ 466,668,158
$ 542,29,236
$ 431,825,988
$ 739,650,466
$ 52,470,668
$ 216,146,977
$ 819,987,18
$ 117,168,530
$ 805,96,715
$ 346,949,466
$ 970,615,88
$ 941,993,340
$ 862,61,35
$ 984,92,344
$ 425,690,689

Dist ← ⍜(≡°√)/+⌵-°⊟
Prep ← (
  ⊜(⊜⋕⊸≠@,)⊸≠@\n
  ⧅<2
  ⊏⍏⊸≡Dist
)

Merge ← (
  ⍜♭₂⊛
  {}
  ⍥(⍣(⊙(°⊂
        □⍥(
          ⟜⊸˜≡⌟(>/+⊃⨂(×₂⋅⧻))
          ◴⊂⊙(♭∩⌟▽⟜¬)
        )∞
      )
      ⊂
    | ∘)
  )∞
  ⊙◌
)

P₁ ← (
  Prep
  # ⍥₁₀(⊙⤙⊡°˜⊂)
  ↙₁₀
  Merge
  ⍆≡◇⧻
  /×↙₋₃
)

P₂ ← (
  Prep
  1_0
  ⊸⍢(
    ⍜°˜⊟(
      ⊙+₁
      ⋅⟜↙
      ⊙(Merge
        ◇⧻⊢)
    )
  | <20⊣)
  -₁⊢
  ⊡
  ×°⊟⊡₀⍉
)

[⊃P₁P₂]
≡(&p $"Part _: _") 1_2