Is egg white lysozyme different from bacterial lysozyme?
PubMed query: "lysozyme"[Title] returns 8673 papers (many of which contain useful data about the activities of bacterial- and chicken-type lysozyme/muramidase against Gram-positive and Gram-negative bacteria).

For example, Gajda &amp; Bugla-Płoskońska (2014, Postepy Hig Med Dosw 21, 1501-1515) write:


  Lysozyme (LZ, muramidase, N-acetylmuramylhydrolase) is a protein occuring in animals, plants, bacteria and viruses.
  
  Lysozyme hydrolyzes the β-1,4 glycosidic bonds between N-acetylmuramic acid (NAM) and N-acetylglucosamine (NAG) of cell
  wall peptidoglycan (PG) in Gram-positive and Gram-negative
  bacteria.
  
  In the animal kingdom, three muramidase types have been
  identified: the c-type (chicken type), the g-type (goose-type) and the
  i-type (invertebrates).
  
  Muramidase shows bactericidal activity mainly against Gram-positive bacteria. Cytolytic activity against cells of Gram-negative bacteria
  has not been proved.


Lysozyme hydrolyzes bacterial cell walls, which is why its muramidase activity is more effective against Gram-positive than Gram-negative bacteria, for the former have thicker cell walls. Whether or not there is any difference in the effectiveness between the bacterial- and chicken-type lysozyme can be found in the papers; i have not read them.

But one of the 8673 papers is titled Hen egg white lysozyme permeabilizes Escherichia coli outer and inner membranes (Derde et al., 2013, J Agric Food Chem. 16: 9922-9929).

The quick-and-dirty answer to the question, is it true that bacterial lysozyme affects Gram-negative bacteria but hen-egg (chicken-type) lysozyme only affects Gram-positive bacteria? is: No.

The 'slow-and-clean' answer can be found in those 8673 papers.