All of these are B.S., but for what it's worth, the one I grew up with:

  Beer before liquor, never been sicker
  Liquor before beer, you're in the clear

While we're at it, the one I knew growing up was probably the most sensible advice and suggested no mixing:

    Grape or grain, but never the twain

Where grain was probably beer (but could've also been Whisky)

O' the demon of rum is abroad in the land. His victims are falling on every hand, The wise and the sinful, the brave and the fair. No station too high for his vengeance to spare. O woman, the sorrow and pain is with you. And so be the joy and the victory too ; With this for your motto and succor divine : The lips that touch liquor shall never touch mine! The lips that touch liquor shall never touch mine!

Sing it with me!

Oh man I wish I knew this, sadly I just know that single line!

To spin a hypothesis into this as to why this is possibly true, think of beer as a walk, wine as a jog and spirits as a sprint. It will take you longer (generally!) to drink the same alcohol content from a pint of beer than it would from a shot of spirits. It can be inferred that one probably consumes alcohol at a faster rate in shot form than in pint form.

It probably helps that beer tends to have a higher water content than liquor.

Ehh, every time i hear this kind og thing beer and wine seems to have switched places

It's supposed to be in order of decreasing ABV, "beer before liquor, never been sicker" is how I know it. The point being, as your inhibitions decrease, your willingness to drink things quickly increases, but lower ABV drinks--especially carbonated ones--have a functional limit on how much can be consumed in a short period of time: stomach capacity.