Cyanobacteria: Gram negative or Gram positive?
Cyanobacteria are clearly didermic (two membranes, inner and outer) bacteria, normally classified as Gram-negative bacteria. If your textbook says several times that they are Gram-positive, instead of a typo, you have a plain mistake.

However, things are not that simple.

According to Hoiczyk and Hansel (2000), this traditional classification of the prokaryotic envelope doesn't work well with cyanobacteria, whose cells show a combination of the features of Gram-positive and Gram-negative bacteria:


  The multilayered peptidoglycan of gram-positive bacteria, with a thickness ranging from 20 to 40 nm, usually forms a physical barrier for the dye. In gram-negative bacteria, however, with their relatively thin sacculus of 2 to 6 nm, the stain can easily be washed out. Together with other characteristics, such as the occurrence of accessory cell wall polymers or the presence of an outer membrane, the peptidoglycan architecture largely determines the different properties of the two cell wall types in terms of mechanical stability, permeability, and resistance toward chemical substances. Although this classification is generally useful, it oversimplifies the concept of bacterial cell walls. It also ignores, for example, that the largest and perhaps most diverse group of bacteria, the cyanobacteria, possess cell envelopes with a combination of these features. (emphasis mine)


Here is an image (ibid.) comparing a Gram-negative cyanobacteria with Escherichia coli:



Electron microscopical comparison of the gram-negative cell envelopes of the cyanobacterium P. uncinatum (A) and E. coli (B). Both bacteria were identically processed using cryosubstitution procedures. Note the combination of gram-positive and gram-negative features present in the cyanobacterial cell wall, like the thick peptidoglycan layer and the outer membrane, respectively. 

Hoiczyk and Hansel (2000) add that...


  ... Despite their overall gram-negative structure, the peptidoglycan layer found in cyanobacteria is considerably thicker than that of most gram-negative bacteria. (emphasis mine)


In conclusion, cyanobacteria are Gram-negative bacteria. However, their envelope show some features os Gram-positive bacteria. Besides that, there is variation withing the group.



Source:


Hoiczyk, E. and Hansel, A. (2000). Cyanobacterial Cell Walls: News from an Unusual Prokaryotic Envelope. Journal of Bacteriology, 182(5), pp.1191-1199.