For "jeannette", might it not be simply a "board", since it is less specific than a sleeveboard?

Finally, you'll need A BOARD that fits inside of the T-shirts you are screening. You'll want one big enough that the T-shirt fits snugly over, so that you can get it straight on the board and not worry about slipping. Put the T-shirt over the board, set it on a table (I recommend this over doing screening on the floor. It's a real pain that way), and position your screen on the shirt. Put ink in the screen and pull the ink across the design to the bottom of the screen, then turn around and pull it back towards the top
http://www.diehippiedie.com/screwball/diyshirt.html

A screen printing device for T-shirts and the like including a flat perforated portion on which the T-shirt is positioned, a vacuum being applied to cause the ink to go through the printing screen.
http://www.freepatentsonline.com/4388862.html

It’s also best to have a hard surface under the shirt to prevent smudges – a wood cutting BOARD was placed inside the shirt for this project
http://www.girlistic.com/diy/tshirt.htm

jeannette – Petite planche étroite et arrondie, montée sur un support, utilisée pour le repassage des pièces que l'on ne peut repasser à plat
[Larousse Lexis]

As for "colle", presumably the fabric has to be glued to the board to stop it moving about under the action of the screen printing squeegee, or might this particular "jeannette" be a more complicated, 3D shape?

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Note added at 43 mins (2007-05-22 10:53:05 GMT)
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GOT IT (one possibility at least) : table adhesive -

The 6 keys to printing with opaque inks:
Print 'off contact' by elevating the screen 5mm.
Keep the squeegee angle upright to print with the sharp edge of the blade
Use slightly more pressure on the squeegee
Keep the design area clear of ink after each print
Increase the number of passes with the squeegee
Use Table Adhesive under your item to hold it flat
http://www.nehoc.com.au/training/is/34/index.htm

'What does Table Adhesive do?' Adhesive boards stop the fabrics from moving when you are printing which eliminated blurring/ smudging and also helps to ...
www.nehoc.com.au/training/faq/prod/htah.htm

Small imprintable items are held in place with liquid table adhesive or adhesive ...
www.swfembroidery.com.au/t-jet.php

Small imprintable items are held in place with liquid table adhesive or adhesive coated ...
www.gumtree.com/london/65/9929665.html

International Coatings 492 Textile Table Adhesive. Product good fo t-shirt printing, piece goods, glass/paper--cloth, art paste up and layouts. ...
www.southwesternprocess.com/miscellaneous_supplies.htm