Why doesn't 1,3-dichloro-2,4-dimethylcyclobutane have a plane of symmetry?
Ivan already stated quite briefly in the comments, that the planes you drew would transform an upward facing chlorine to a downward facing chlorine and vice versa, and that the same holds for the methyl groups. This is of course true and the first and most obvious point, why there is no symmetry plane. 

A more close investigation will, however, point out another important feature of the compound itself. The cyclobutane ring itself is not planar. In the (electronically) lowest conformation of the molecule it has no symmetry whatsoever (technically $C_1$ symmetry), not even inversion symmetry like your book states. This is due to the fact, that the four-membered ring is puckered. This is best visualised by looking at the 3D structure of the compound. There are multiple programs that will allow you to view such a structure and I will xyz coordinates at the end of this posting.

The following image will hopefully already show you what I am talking about. It was optimised on the DF-BP86/def2-SVP level of theory.



In order for the the molecule to have inversion symmetry, the shown angle needs to be exactly $0^\circ$. The following animation shows the molecule rotating hopefully further clarifying this point.



You can clearly see the puckering of the cyclobutane ring.

One final comment: For a molecule like this it really is necessary to specify the stereochemistry with the name. There are other stereoisomers, which do indeed have a symmetry plane. I leave it to you to figure out which of them do.




   Except the last one, all of them have at least one plane of symmetry.




Addendum: Transition state structure

The symmetric ($C_\mathrm{i}$) transition state is on the DF-BP86/def2-SVP level of theory only about $\Delta G (298.15~\mathrm{K}) = 8~\mathrm{kJ/mol}$ higher in energy than the puckered asymmetric forms. It is therefore worthwhile to have a quick look at its structure. In the following graphics the red ball indicates the inversion centre.
Below that is an animation of the transitional mode. It briefly pauses at the puckered structures (these are extrapolated, not optimised structures) and the transition state to make it easier to see the movement in between them.
I would expect this process to be readily available at room temperature and therefore almost impossible to distinguish the different lower lying structures.






Coordinates of the lowest structure:

18
G09D.01; E(RB-P86/def2-SVP/W06) = -1154.66760674 A.U.
C        0.051334000      1.246599000      0.089432000
C       -0.768087000      0.117382000      0.789824000
C        0.002884000     -0.950933000     -0.020424000
C        1.086640000      0.142749000     -0.293350000
H        0.449081000      2.008047000      0.790972000
H       -0.535742000      0.060683000      1.871003000
H       -0.532992000     -1.105843000     -0.982729000
H        1.517474000      0.183811000     -1.312785000
C        0.343625000     -2.291820000      0.610089000
H        0.939420000     -2.156389000      1.535300000
H       -0.586821000     -2.839917000      0.866159000
H        0.932752000     -2.926935000     -0.083624000
C       -0.606103000      1.926898000     -1.110977000
H       -1.475939000      2.532182000     -0.784546000
H        0.116584000      2.600709000     -1.616245000
H       -0.977177000      1.195914000     -1.858831000
Cl       2.502295000      0.083512000      0.845211000
Cl      -2.570729000      0.126790000      0.662589000




Coordinates of the transition state (X is a dummy atom indicating the inversion centre)

19
G09D.01; E(RB-P86/def2-SVP/W06) = -1154.6659061600 A.U.
C       -0.188954000      1.025076554      0.415795307
C       -0.873066584     -0.383572154      0.501227861
C        0.188954000     -0.991627446     -0.457606693
C        0.898153416      0.404477846     -0.509590139
H        0.242239584      1.289691385      1.404629307
H       -0.864241753     -0.820445015      1.518694431
H       -0.267326416     -1.206068615     -1.446440693
H        0.939502247      0.849712985     -1.522875569
C        0.953473154     -2.212350139      0.042720768
H        1.426842139     -2.018382400      1.026791199
H        0.267219307     -3.079692431      0.145048491
H        1.760843738     -2.494108569     -0.663022216
C       -0.974378846      2.203987861     -0.147249232
H       -1.806836262      2.489927431      0.529225784
H       -0.309030693      3.083873569     -0.266301509
H       -1.418479861      1.964027600     -1.135500801
Cl       2.591747000      0.489781723      0.126797539
Cl      -2.591747000     -0.506506277     -0.064080461
X        0.006271708      0.013588700     -0.012543416