What do those square buttons say?

[figheadair]

If you don't mind me asking, do you know the origin of these buttons? Where do they come from and why did they become so ubiquitous on certain kilt jackets?

I'm afraid that I don't know their history but the earliest use I've seen is the capbadge used by a number of Volunteer Regiments in the 1860's including the:

• 1st City of Edinburgh Rifle Volunteers
  Highland Company Badge For The Queens Edinburgh Rifles

It seems likely that its use was later adopted for civilian Highland Dress, possibly as late as the early-mid 20th century.

Do you have the older version that includes the "stand in friendship" part?

Or are those concepts implied by the "shoulders together" phrase?

The ‘in friendship’ element is implied.

The term appears to have come from the Gaelic song Ri Guaillibh a Chèile by Donnchadh Mac Ille Ruaidh, born in Kintyre in 1849. The chorus of the song is:

Clanna nan Gàidheal ri guaillibh a chèile
Faicibh a' tighinn iad 's faram nan ceuman;
Bratach na Gàidhlig tha nis air a sgaoileadh,
Clanna nan Gàidheal ri guaillibh a chèile.

Clanna nan Gàidheal ri guaillibh a chéile meaning Gaels standing shoulder to shoulder.

The Gaelic Society of Inverness used the motto Clanna nan Gàidheal ri guaillibh a chéile from the 1870’s.

I’d love to know the history behind kilt jackets going from thistle buttons and round buttons to these square buttons.

I don’t know but have a feeling that the square version is a mid-20th century mass production fashion that has become popular via the hire trade.